Frankly, the judges I practice in front of just don't allow Alford pleas or No Contest pleas. Judges really want to hear a client say "I did it." And, again and again, judges say, "If you can't tell me what you did, you should just go to trial."
I can think of only one time when a judge not only allowed me to enter an Alford plea, he encouraged it. It was in night court. And my client was a little old woman, who was completely deaf. As in old-people-deaf, not knows-sign-language-deaf. I can't remember now what the little old lady was accused of, but I think everyone in the courtroom was just appalled that she had been kept in lockup overnight.
I don't remember a whole lot about the case, I think maybe the prosecutor was offering some kind of non-criminal disposition (no criminal record) with time served. But I remember explaining to the old lady again and again that if she could say "she did it" she could just go home. Again and again, the old lady would say, "Ok, I'll do it," and then, I'd ask her, shouting, "Mrs. Old Lady, did you do what they say you did?" and she was said, "No sweetie." And I would try again to explain it to her again, but I would have to shout in the courtroom, "Remember what I said, you could go home?" And she would say, "Oh, yes, that's right, okay," and then we go through the whole thing again.
Finally, the judge, who was pretty cool, shouted, "I'll allow an Alford Plea. Mrs. Old Lady, would you like to go home?" The old lady said yes.
"Alright, that's sufficient," the judge said.
Overall, though, I would say that judges want to hear that guilty plea. Why? Maybe because it makes it harder for that plea to come back on an appeal. A defendant with pleaders remorse will have a harder time saying "I didn't know what I was doing" if they already admitted their criminal act.
And, I have to say, I think there is some truth to Manitor's comment:
The system demands admitting guilt. That's why it has evolved into the efficient submission-extraction process it is today.
What really matters is not punishing crime; what matters is repeating the ritual of having individuals submit their autonomous will to the authority of the state by admitting guilt. Honesty and truth have nothing to do with it.
What about everyone else? Do your judges allow Alford or N.C. pleas? It seems that you hear about them sometimes on television with regard to celebrity cases, so I wonder if that's because they're popular elsewhere or they're popular in celebrity cases.